In this paper, we present some major algorithmic improvements to fast correlation attacks. In previous articles about fast correlations, algorithmics never was the main topic. Instead, the authors of these articles were usually addressing theoretical issues in order to get better attacks. This viewpoint has produced a long sequence of increasingly successful attacks against stream ciphers, which share a main common point: the need to find and evaluate parity-checks for the underlying linear feedback shift register. In the present work, we deliberately take a different point of view and we focus on the search for efficient algorithms for finding and evaluating parity-checks. We show that the simple algorithmic techniques that are usually used to perform these steps can be replaced by algorithms with better asymptotic complexity using more advanced algorithmic techniques. In practice, these new algorithms yield large improvements on the efficiency of fast correlation attacks.

From the representation of Boolean functions based on the Cayley graph adjacency matrix, we evaluate, for each Boolean function, the product of all the values of his Walsh spectrum. An application to the extremal balanced Boolean functions is given.

The subject of this paper is the algebraic study of the adjacency matrix of the Cayley graph of a Boolean function. From the characteristic polynomial of this adjacency matrix we deduce its minimal polynomial.

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We prove new upper bounds for the covering radii ρ(n) and ρ B (n) of the first order Reed-Muller code R(1, n). Although these bounds be actually theoretical, they improve the classical Helleseth-Kløve-Mykkeltveit (H.K.M.) bound 2 n−1 − 2 n 2 −1 .